9. The Valley of the Mississippi, from Cape Girardeau to the delta, is estimated to contain 16,000 square miles, of 150 feet deep; it therefore contains 66,908,160,000,000 cubic feet, or 454 cubic miles.

Prof. GERMAIN gave some views of his connected with this subject.

On motion of Prof. W. R. JOHNSON, it was recommended to the Association that thanks be presented to the Committee for the very able manner in which they have discharged their duty in this matter, and that the report be published in full as early as practicable.

Dr. R. COATES moved that this Section adjourn to meet to-morrow at 10 A. M., which was agreed to.

September 22.

R. W. GIBBES, Sec'ry.

SECTION OF GENERAL PHYSICS, &c.

Second Meeting.

The Section met at 9. A. M. The following papers were read:

ON THE FUNDAMENTAL PRINCIPLES OF MATHEMATICS. BY PROF. STEPHEN ALEXANDER.

Prof. ALEXANDER remarked, the object of all scientific research was truth; a term too valuable to be misunderstood, and yet too general to admit of a ready definition. He proceeded, however, to characterize it, in some of its various aspects, observing that while each is applicable to its own object, that is true in mathematics, which, under the existing system of things, is supposable; that is true in physics, which, under the existing system of things, has been permitted to exist; that is true in matters of taste which is consistent with the laws of beauty, deduced from the observation of things actual; and that is true in morals, (in the highest and best sense in which it is good,) which is consistent with what is found in the GREAT SOURCE OF ALL GOOD.

He next proceeded to state that mathematics had not to do with things, but the relations of things, and it was sufficient that those relations should be supposable; and that the certainty of mathematical

reasoning rested upon the fact, that those relations could be more readily understood and completely defined, than the properties of the things themselves.

He stated, moreover, that these were constituted relations, and not mere figments of the human mind; the things which we had to deal with, being made, in certain respects, not merely what they were, but as they were. Thus, that two bodies did not occupy the same space, and that it was true that one day of the week must follow another, were not true because his audience and himself might think so, but because the Creator had made them so.

He next commented upon the general term which was used to designate that to which mathematical reasoning was applicable; viz. quantity; and said, that in so far as mathematics had to do with it, it was that which admitted of the distinction of greater or less. Moreover, quantities were of the same species when each in itself exceeded its less in the self same respect in which the other in itself exceeded its less; whatever might be true of the boundaries or limits of either. Thus, a straight line and a curve were of the same species, since each exceeded its less in length; so, also, an hour and a minute were of the same species, since each exceeded its less in duration.

He remarked, that the only point of resemblance between quantities of different species, was to be found in the fact, that the distinction of greater and less was admissible in the case of every species; and hence it was possible to compare the ratio of two quantities of one species, with that of two quantities of another species.

He proceeded to the more special consideration of the two great relations of things, time, and space; remarking, that space might be described, as that wherein there was room for the separate existence of material substances; and duration, as that wherein there was room in another sense for the separate, and therefore successive, occurrence of events.

He next commented upon the nature of zero; showing that it implied the absence of the species of quantity which happened to be the subject of investigation, and not the absence of every other. That, thus, the surface which bounded a solid quantity, was not somewhat in the same sense in which the solid was somewhat, viz. in the property of occupying space, but only somewhere, viz. where the solid terminated, and space met it; the space without met it, though that surface was still somewhat in superficial extent. That the line which bordered the surface was not somewhat in this last respect, but only somewhere; though still somewhat in length.

Lastly, that the point which terminated the line was not somewhat in any respect, but only somewhere; viz. at the end of the line; and that the same was true when a point was otherwise situated; e. g. the centre of a sphere.

He remarked, that an instant also existed as the limit of duration; e. g. the midnight with which one day terminated and the other began; but this existed not where the one ended and the other began, but when; or such a limit was not somewhere, but, if there were such a word, somewhen. That rest, or the zero of motion, existed when and where a body came to rest, and that shadow existed when and where light was absent.

He moreover considered the subject of infinity, and distinguished three sorts of infinity.

He remarked that he should designate a quantity as absolutely infinite, if it were so great as to be destitute of any boundary or limit; and gave the only two recognised examples of this, viz. boundless space, and that duration which is made up of ETERNITY, PAST and FUTURE. Eternity past was that which found its realization in the Divine Pre-existence, and Eternity future was to be found in the endless duration of the same; and nothing less than the combination of both of these, nothing short of it, constituted the absolute infinity of duration.

He moreover remarked that he should designate a quantity as being specifically infinite, if it were just as boundless as those last described, but in certain respects only. He gave as examples :

1. A straight line without termination in either direction from a point which might be assumed in that line, such a line would be specifically infinite; viz. in length.

2. A surface without border which would be specifically infinite; viz. in length, breadth, and superficial area. He drew the conclusion, moreover, that an interminable line which was not straight throughout, must be longer than that which was perfectly straight, since the former not merely extended through space in its length, but intruded somewhat upon the breadth of space.

He next remarked that he should designate a quantity as being in comparison with another, relatively infinite, if its ratio to that other were too great to be estimated; that in this sense alone could we speak of an infinite number of things, or of an infinitely great number in the abstract. The like must be true of velocity, and also of mere mechanical force.

He next considered the subject of motion as applicable to mathe

H

matical quantities, and gave some illustrations showing, that when bodies moved they forsook the positions in space which they at first occupied, and that the position occupied by the centre of gravity, or any specified point of reference with regard to the body, was in like manner left behind, and a new position in space be so situated, as to be the centre of gravity, or point of reference of the body; both the space first occupied and the positions left behind having, themselves, no motion. He therefore designated the motion of a mathematical point, as being a pleasant fiction, and said that, were it otherwise, a point, which was nothing, might, by motion, produce a line which had length.

He next supposed a point (P) to be assumed in an interminable line, and remarked, that all that portion of the line on the one side of the point, must be regarded as being in effect the half of the line, and all on the other side as being, in effect, the other half. But if a new point (P') were assumed in the same line at any finite distance from

the other, the two portions, one on the one side, and the other on the other side, must, as before, be regarded as being, in effect, the halves of the line; though all the intervening portion of the line (P P') had (at the new point of division) been taken from the one half, and added to the other. Hence, any finite straight line must be regarded as good for nothing, in comparison with a straight line interminable in only one direction; or if the line thus interminable were used as the measuring unit, its ratio to any finite straight line must be represented by. Any other finite straight line, however great or however small, must in like manner be represented by zero in comparison with the same measuring unit; and the ratio of the one finite quantity to the other, be therefore represented by . Hence was a symbol of indeterminateness. In this case that indeterminateness would be absolute. Prof. A. also remarked with regard to another common case, in P (X—a)TM which which the value of % might enter; viz. Q (X-a)",

when

X = a; that, in this case, the numerator and denominator both were reduced to zero, because the multiplier in each case vanished, so that no process of multiplication was possible; and there was, in each case, absolutely no result: insomuch, that vanishing fractions might, in this point of view, be rather termed vanished fractions.

[Prof. A. next, incidentally, spoke of the reason why the radius in

the investigation, of analytic trigonometry must be regarded as positive when measured from the centre outwards to any point on the circumference.

He adverted for this purpose to the method employed for the determination of the position of a point in space, showing that the distance from the origin in one direction must be regarded as positive, to whichever of the three axes reference was employed, and that a negative distance could, in every case, be obtained, by measuring backward from (P) the farther extremity of the opposite distance (PM), an extent (P P') greater than the positive distance, and thus passing in the opposite direction to the other side of the origin.

He next supposed the angles made by some (or all) of the axes with each other, to be so increased that those axes should all be brought into the same plane. The three directions from the origin outward would thus be found to be positive, while the opposite directions must be regarded as negative. As, moreover, any number of groups of axes, three in each, might thus be clustered around the same origin, all directions from the centre outward must then be regarded as positive, and the contrary negative.]

Prof. ALEXANDER then resumed the comparison of finite quantities with the infinite, and in manner as before, proceeded to show that if a plane were supposed to extend through all space, all the portion on the one side of this plane must be regarded as being, in effect, the half of all space, and that all on the other side, as being, in effect, the other half. The like would, however, be true if another such plane were to extend through space parallel to the first; though to what before constituted the one half, would be added all the space between the two planes, and the same subtracted from the other. Hence, reasoning in the manner as before, we must conclude that this intervening space, though boundless in some of its dimensions, must be regarded as good for nothing in comparison with the half of all space; i. e. the half in the sense already described.

For like reasons, any finite portion of time must be regarded as nothing, in comparison with either eternity past or eternity future; and thus we might, in some humble measure, discern how, in view of a mind which could grasp the whole, "a thousand years" would be "as one day, and one day as a thousand years."

He lastly considered the question-whether, if the visible creation were annihilated, space would still exist, and concluded that we had not sufficiently accurate ideas of such a state of things to determine with regard to it; but insisted that, in any event, space could not